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| In [[theoretical physics]], the '''Bogoliubov transformation''', named after [[Nikolay Bogolyubov]], is a [[unitary transformation]] {{Dubious|date=July 2009}} from a [[unitary representation]] of some [[canonical commutation relation algebra]] or [[canonical anticommutation relation algebra]] into another unitary representation, induced by an [[isomorphism]] of the commutation relation algebra. The Bogoliubov transformation is often used to diagonalize [[Hamiltonian (quantum mechanics)|Hamiltonian]]s, which yields the steady-state solutions of the corresponding [[Schrödinger equation]]. The solutions of [[BCS theory]] in a homogeneous system, for example, are found using a Bogoliubov transformation. The Bogoliubov transformation is also important for understanding the [[Unruh effect]], [[Hawking radiation]] and many other topics.
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| == Single bosonic mode example ==
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| Consider the canonical [[Commutator|commutation relation]] for [[bosonic]] [[creation and annihilation operators]] in the harmonic basis
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| :<math>\left [ \hat{a}, \hat{a}^\dagger \right ] = 1</math> | |
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| Define a new pair of operators
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| :<math>\hat{b} = u \hat{a} + v \hat{a}^\dagger</math>
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| :<math>\hat{b}^\dagger = u^* \hat{a}^\dagger + v^* \hat{a}</math>
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| where the latter is the [[hermitian conjugate]] of the first. The Bogoliubov transformation is a canonical transformation of these operators. To find the conditions on the constants ''u'' and ''v'' such that the transformation remains canonical, the commutator is expanded, viz.
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| :<math>\left [ \hat{b}, \hat{b}^\dagger \right ] | |
| = \left [ u \hat{a} + v \hat{a}^\dagger , u^* \hat{a}^\dagger + v^* \hat{a} \right ]
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| = \cdots = \left ( |u|^2 - |v|^2 \right ) \left [ \hat{a}, \hat{a}^\dagger \right ]. </math>
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| It can be seen that <math>\,|u|^2 - |v|^2 = 1</math> is the condition for which the transformation is canonical. Since the form of this condition is reminiscent of the [[Hyperbolic function|hyperbolic identity]] <math>\cosh^2 x - \sinh^2 x = 1</math>, the constants ''u'' and ''v'' can be parametrized as
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| :<math>u = e^{i \theta_1} \cosh r \,\!</math>
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| :<math>v = e^{i \theta_2} \sinh r \,\! .</math>
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| ===Applications===
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| The most prominent application is by [[Nikolai Bogoliubov]] himself in the context of [[superfluidity]].<ref>[[Nikolai Bogoliubov]]: ''On the theory of superfluidity'', J. Phys. (USSR), 11, p. 23 (1947)</ref> Other applications comprise [[Hamiltonian (quantum mechanics)|Hamiltonians]] and excitations in the theory of [[antiferromagnetism]].<ref name="Kittel">See e.g. the textbook by [[Charles Kittel]]: ''Quantum theory of solids'', New York, Wiley 1987.</ref> When calculating quantum field theory in curved space-times the definition of the vacuum changes and a Bogoliubov transformation between these different vacua is possible, this is used in the derivation of [[Hawking radiation]].
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| == Fermionic mode ==
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| For the [[Commutator|anticommutation]] relation
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| :<math>\left\{ \hat{a}, \hat{a}^\dagger \right\} = 1</math>,
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| the same transformation with ''u'' and ''v'' becomes | |
| :<math>\left\{ \hat{b}, \hat{b}^\dagger \right\} = (|u|^2 + |v|^2) \left\{ \hat{a}, \hat{a}^\dagger \right\}</math>
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| To make the transformation canonical, ''u'' and ''v'' can be parameterized as
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| :<math>u = e^{i \theta_1} \cos r \,\!</math> | |
| :<math>v = e^{i \theta_2} \sin r \,\! .</math>
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| ===Applications===
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| The most prominent application is again by Nikolai Bogoliubov himself, this time for the [[BCS theory]] of [[superconductivity]] .<ref name="Kittel" /> The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite <math>\,\langle a_i^+a_j^+\rangle</math>-terms, i.e. one must go beyond the usual [[Hartree-Fock method]] (-> [[Hartree-Fock-Bogoliubov method]]). Also in [[nuclear physics]] this method is applicable since it may describe the "pairing energy" of nucleons in a heavy element.<ref>[[Vilen Mitrovanovich Strutinsky]]: ''Shell effects in nuclear physics and deformation energies'', Nuclear Physics A, Vol. 95, p. 420-442 (1967), [http://www.sciencedirect.com/science/article/pii/0375947467905106] .</ref>
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| == Multimode example ==
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| The [[Hilbert space]] under consideration is equipped with these operators, and henceforth describes a higher-dimensional [[quantum harmonic oscillator]] (usually an infinite-dimensional one).
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| The [[ground state]] of the corresponding [[Hamiltonian (quantum mechanics)|Hamiltonian]] is annihilated by all the annihilation operators:
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| :<math>\forall i \qquad a_i |0\rangle = 0</math>
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| All excited states are obtained as [[linear combination]]s of the ground state excited by some creation operators:
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| :<math>\prod_{k=1}^n a_{i_k}^\dagger |0\rangle</math>
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| One may redefine the creation and the annihilation operators by a linear redefinition:
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| :<math>a'_i = \sum_j (u_{ij} a_j + v_{ij} a^\dagger_j)</math>
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| where the coefficients <math>\,u_{ij},v_{ij}</math> must satisfy certain rules to guarantee that the annihilation operators and the creation operators <math>a^{\prime\dagger}_i</math>, defined by the [[Hermitian conjugate]] equation, have the same [[commutator]]s
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| for bosons and anticommutators for fermions.
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| The equation above defines the Bogoliubov transformation of the operators.
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| The ground state annihilated by all <math>a'_{i}</math> is different from the original ground state <math>|0\rangle</math> and they can be viewed as the Bogoliubov transformations of one another using the [[operator-state correspondence]]. They can also be defined as [[squeezed coherent state]]s. BCS wave function is an example of squeezed coherent state of fermions.<ref>Svozil, K. (1990), "Squeezed Fermion states", ''Phys. Rev. Lett.'' '''65''', 3341-3343. {{doi|10.1103/PhysRevLett.65.3341}}</ref>
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| == References ==
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| {{More footnotes|date=February 2008}}
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| {{reflist}}
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| ==Literature==
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| The whole topic, and a lot of definite applications, are treated in the following textbooks:
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| * J.-P. Blaizot and G. Ripka: Quantum Theory of Finite Systems, MIT Press (1985)
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| * A. Fetter and J. Walecka: Quantum Theory of Many-Particle Systems, Dover (2003)
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| * Ch. Kittel: Quantum theory of solids, Wiley (1987)
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| ==External links==
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| {{DEFAULTSORT:Bogoliubov Transformation}}
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| [[Category:Quantum field theory]]
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| [[Category:Canonical unitary transformation]]
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The child of your , is aware perseverance and perseverance are important elements in terms of an effective job- . His first album, Keep Me, generated the Top hits “All My Girlfriends “Country and Say” Person,” luke bryan my kind of night tour when his hard work, Doin’ Thing, discovered the vocalist-about three straight No. 3 single people: In addition Contacting Is a Fantastic Point.”
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